Question

Prove the diagonals of the square with vertices P (0,4), Q (4,4), R (0,0) and S (4,0) are perpendicular bisectors of each other.

Answer 1

Answer: The slope of diagonal PS is -1. The The slope of diagonal QR is 1. The midpoint of PS is (2,2). Midpoint of QR is (2,2).

Explanation:

The given vertices are P (0,4), Q (4,4), R (0,0) and S (4,0).

If a line passing through the point $(x_1,y_1)$ and $(x_2,y_2)$, then the slope of the line is,

$m=\frac{y_2-y_1}{x_2-x_1}$

The slope of diagonal PS is,

$m_1=\frac{0-4}{4-0}= \frac{-4}{4} =-1$

The slope of diagonal QR is,

$m_2=\frac{0-4}{0-4}= \frac{-4}{-4} =1$

The slope of both diagonals are 1 and -1, therefore the product of their slopes is -1. So, the diagonals are perpendicular to each other.

The formula for midpoint is,

$midpoint=(\frac{x_1+x_2}{2} ,\frac{y_1+y_2}{2} )$

The midpoint of PS is,

$\text{midpoint of PS}=(\frac{0+4}{2} ,\frac{4+0}{2})=(2,2)$

The midpoint of QR is,

$\text{midpoint of QR}=(\frac{4+0}{2} ,\frac{4+0}{2})=(2,2)$

The midpoint of both diagonals are same, therefore diagonals are bisecting each other.

Therefore, the slope of diagonal PS is -1. The The slope of diagonal QR is 1. The midpoint of PS is (2,2). Midpoint of QR is (2,2).

Answer 2

-1
1
(2,2)
(2,2)
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